An integral representation of the modified Bessel function is given as $$k_v = \frac{1}{2}\int_{-\infty}^\infty e^{-x\cosh t- vt} \, dt.$$
How do I find the leading order behavior of $k_{ip}$, as $x\to\infty$ and $p/x\to \infty$.
I suppose I have to use the method of steepest descent for which the saddle points are satisfy -
$\sinh(t)= \frac{ip}{x}$ , which gives me $t= \ln\left( i\frac{p}{x} +\frac{\sqrt{x^2-p^2}}{x} \right)$ $= i \tan^{-1}\left(\frac{p}{\sqrt{x^2-p^2}} \right) + 2ni\pi.$ Now I don't know how to proceed further, also since there are infinite saddle points, how to determine which of them actually contributes to the integral? Thanks in advance.